The Ground State Energy
of Relativistic One-Electron Atoms
According to Jansen and Heß
Raymond Brummelhuis, Heinz Siedentop, and Edgardo Stockmeyer
Received: August 16, 2001 Revised: June 15, 2002
Communicated by Alfred K. Louis
Abstract. Jansen and Heß – correcting an earlier paper of Dou-glas and Kroll – have derived a (pseudo-)relativistic energy expres-sion which is very successful in describing heavy atoms. It is an ap-proximate no-pair Hamiltonian in the Furry picture. We show that their energy in the one-particle Coulomb case, and thus the resulting self-adjoint Hamiltonian and its spectrum, is bounded from below for
αZ≤1.006.
2000 Mathematics Subject Classification: 81Q10, 81V45
1 Introduction
The energy of relativistic electrons in the electric field of a nucleus of charge
Zeis described by the Dirac Operator
Dγ =cα· ~
i∇+mc
2β
− γ
|x| (1)
to define electrons directly by their external field (Furry picture). (See Sucher [17] for a review.) This strategy, however, meets immediate difficulties, since the projectionχ(0,∞)(Dγ) is much harder to find for positiveγthan forγ= 0.
To handle this problem Douglas and Kroll [4] used an approximate Foldy-Wout-huysen transform to decouple the positive and negative spectral subspaces of
Dγ. Their approximation is perturbative of second order in the coupling con-stant γ. Jansen and Heß [11] — correcting a sign mistake in [4] — wrote down pseudo-relativistic one- and multi-particle operators to describe the en-ergy which were successfully used to describe heavy relativistic atoms (see, e.g., [12]).
This derivation yields the operator (see [11], Equation (17))
HDext=βe+E+
1
2[W, O], (2)
where
e(p) := pp2+m2, (3)
E := A(V +RV R)A, (4)
O := βA[R, V]A, (5)
A(p) := µ
e(p) +m
2e(p) ¶12
, (6)
R(p) := α·p
e(p) +m, (7)
W(p,p′
) = β O(p,p ′)
e(p) +e(p′). (8)
(Note that we writepfor|p|.) HereV is the external potential which in the case at hand is the Coulomb potential, and in configuration space it is multiplication by−γ/|x|.
This operator – which acts on four spinors – is then sandwiched by the projec-tion onto the first two components, namely (1 +β)/2. The resulting upper left corner matrix operatorJγ :C∞
0 (R3)⊗C2→L2(R3)⊗C2 is
Jγ :=Bγ+γ2K˜ =e−(γ/(2π2))K+γ2K.˜ (9)
with
K(p,p′
) = (e(p) +m)(e(p
′) +m) + (
p·σ)(p′·σ)
n(p)|p−p′
|2n(p′) (10)
where n(p) := (2e(p)(e(p) +m))1/2, i.e., Bγ is the Brown-Ravenhall operator [2]. (See also Bethe and Salpeter[1] and Evans et al. [5]).
The last summand in (9) is given by the kernel
˜
K(p,p′
) =−12
Z
dp′′
[W(p,p′′
)P(p′′ ,p′
) +P(p,p′′
)W(p′′ ,p′
with
P(p,p′
) = σ·p(e(p
′) +m)
−(e(p) +m)σ· gp′ 2π2n(p)|p−p′
|2n(p′) (12)
and
W(p,p′
) = P(p,p
′
)
e(p) +e(p′). (13)
Introducing b(p) := p/n(p) and a(p) := ((e(p) +m)/2e(p))1/2 we get more explicitly
˜
K(p,p′
)
= 1
2(2π2)2 Z
dp′′ 1
|p−p′′|2|p′′−p′|2
µ 1
e(p) +e(p′′)+
1
e(p′′) +e(p′)
¶
£
(ωp·σ) (ωp′·σ)b(p)a(p′′)2b(p′)−(ωp·σ) (ωp′′·σ)b(p)b(p′′)a(p′′)a(p′) +a(p)b(p′′
)2a(p′
)−(ωp′′·σ) (ω
p′·σ)a(p)b(p′′)a(p′′)b(p′)¤. (14) (For later use we name the expression in the first line of the integrand in (14)
C and the four terms in the square bracketT1, ..., T4.) The corresponding energy in a stateu∈C∞
0 (R3)⊗C2is
J(u) := (u, Jγu) =B(u) +γ2(u,Ku˜ ) (15) with
B(u) = Z
R3
dpe(p)|u(p)|2− γ 2π2
Z
R3
dp
Z
R3
dp′ u(p)∗
K(p,p′
)u(p′
) (16)
It is the quadratic formJ which is our prime interest.
Throughout the paper we will use the following constants γc := 4π(π2+ 4− √
−π4+ 24π2−16)/(π2−4)2),γB
c := 2/(π/2+2/π), anddγ := 1−γ−4 √
2(3+
√
2)γ2. Our goal is to show
Theorem 1. For all nonnegative masses mthe following holds:
1. If γ ∈[0, γc] then J is bounded from bellow, i.e., there exist a constant c∈Rsuch that for all u∈C∞
0 (R3)⊗C2
J(u)≥ −cmkuk2.
2. Ifγ > γc, thenJ(u)is unbounded from below. 3. Ifγ∈[0, γB
c ] then
J(u)≥dγmkuk2.
Note that γc ≈ 1.006077340. Because γ = αZ where α is the Sommerfeld fine structure constant which has the physical value of about 1/137 and Z
It also means that the method is applicable for all αZ where the Coulomb-Dirac operator can be defined in a natural way through form methods (Nenciu [15]). — Note, in particular, that the energy is bounded from below, even if
γc > γ > 1 although the perturbative derivation of the symmetric operator
Hext
D is questionable in this case.
We would like to remark that the lower bound can most likely be improved for positive masses. In fact, we conjecture that the energy is positive for all sub-critical γ. However, this is outside the scope of this work.
According to Friedrichs our theorem has the following immediate consequence: Corollary 1. The symmetric operatorJγ has a unique self-adjoint extension whose form domain containsC∞
0 (R3)⊗C2 forγ∈[0, γc].
In fact for γ < γc, since the potential turns out to be form bounded with relative bound less than one, the self-adjoint operator defined has form domain
H1/2(R3)⊗C2.
The structure of the paper is as follow: in Section 2 using spherical symme-try we decompose the operator in angular momentum channels. In Section 3 we prove the positivity of the massless operators. Since these operators are homogeneous under dilation an obvious tool to use is the Mellin transform, a method that previously has been used with success to obtain tight estimates on critical coupling constant (see, e.g., [3]). In Section 4 we find that the differ-ence between the massless and the massive operator is bounded. Finally, some useful identities are given in the Appendix.
2 Partial Wave Analysis of the Energy
To obtain a sharp estimate for the potential energy we decompose the operator as direct sum on invariant subspaces. Because of the rotational symmetry of the problem one might suspect that the angular momenta are conserved quantities. Indeed, as a somewhat lengthy calculation shows, the total angular momentum
J=1
2(x×p+σ) commutes withH
ext. In fact we can largely follow a strategy carried out by Hardekopf and Sucher [9] and Evans et al. [5] in somewhat simpler contexts.
We begin by observing that those of the spherical spinors
Ωl,m,s(ω) :=
q
l+s+m
2(l+s)Yl,m−1
2 (ω) q
l+s−m
2(l+s)Yl,m+12(ω)
s= 1 2
−ql+s−m+1 2(l+s)+2Yl,m−1
2 (ω) q
l+s+m+1
2(l+s)+2Yl,m+12(ω)
s=− 1 2
(17)
with l = 0,1,2, ... and m = −l − 1 2, ..., l+
1
on the unit sphereS2(see, e.g., [14], p. 421) with the convention thatYl,k= 0, be written as
u(p) = X
We now remind the reader that the expansion of the Coulomb potential in spherical harmonics is given by
1
where the Pl are Legendre polynomials. [See Stegun [16] for the notation and some properties of these special functions.]
Inserting the expansion (18) and (19) into (15) yields
The Legendre functions of the second kind appear here for exactly the same reasons as in the treatment of the Schr¨odinger equation for the hydrogen atom in momentum space (Fl¨ugge [6], Problem 77).] To obtain (21), we also use that (ωp·σ)Ωl,m,s(ωp) =−Ωl+2s,m,−s(ωp) (see, e.g., Greiner [8], p. 171, (12)). The
operatorshl,sdefined by the sesquilinear form (21) via the equation (f, hl,sf) =
Jl,s(f) are reducing the operatorHexton the corresponding angular momentum subspaces.
3 The Massless Operators and Their Positivity
To proceed, we will first consider the massless operators. The lower bound in the massive case will be a corollary of the positivity of the massless one. The energy in angular momentum channel (l, m, s) in the massless case can be read of from (14) and is given by
Using the simplifications of Appendix A, Formulae (57) and (59) we get
˜
Since the operator in question is homogeneous of degree minus one we Mellin transform (see Appendix B) the quadratic formεl,s. If we write this form as a functionalJl,s# of the Mellin transformed radial functions f#, we get
whereBl,s# is the Brown-Ravenhall energy in angular momentum channel (l, s)
in Mellin space, i.e.,
Bl,s#(g) :=
Z ∞
−∞
dt|g(t+i/2)|2h1−γ2(Vl(t) +Vl+2s(t))
i
(29)
with
Vl(t) = r
2
πq
#
l (t−i/2) =
1 2 ¯ ¯ ¯ ¯ ¯
Γ¡l+1−it
2 ¢
Γ¡l+2−it
2 ¢
¯ ¯ ¯ ¯ ¯ 2
(30)
(see Tix [19] [note also the factor p2/π which is different from Tix’s original formula]) and
F#(t) =√2π³ql#(t−i/2)−q#l+2s(t−i/2)´2. (31)
Formulae (29), (30), and (31) are obtained from (24), (25), and (27) using the fact that the occurring integrals can be read as a Mellin convolution which is turned by the Mellin transform into a product (see Appendix B, Formulae (61) and (63)).
Note that Vl is the Coulomb potential after Fourier transform, partial wave analysis, and Mellin transform.
3.1 Positivity of the Brown-Ravenhall Energy To warm up for the minimization ofJl,s# we start withB
#
l,s only. To this end
we first note
Lemma 1. We have
Vl+1(t)≤Vl+1(0)≤Vl(0). (32) Note, that this is similar to Lemma 2 in [5].
Proof. First note thatq0≥q1≥q2...which follows from the integral represen-tation in [21], Chapter XV, Section 32, p. 334. This implies
¯ ¯ ¯q
#
l+1(t−i/2) ¯ ¯ ¯=
1
√
2π
¯ ¯ ¯ ¯
Z ∞
0
ql+1(p)p−itdp p
¯ ¯ ¯ ¯≤
1
√
2π
Z ∞
0
ql+1(p)dp
p
≤√1
2π
Z ∞
0
ql(p)dp
p ,
(33)
which implies the lemma.
Theorem 2. For allu∈C∞
0 (R3)⊗C2 andm= 0 we haveB ≥0 if and only
Proof. Note that
Vl(t) +Vl+2s(t)≤V0(0) +V1(0) = π
2 + 2
π. (34)
Thus
B#l,s(g)≥
Z ∞
−∞
dt|g(t+i/2)|2
µ 1−γ2
µπ
2 + 2
π
¶¶
(35)
which implies that the energy is nonnegative ifγ≤2/(π/2 + 2/π).
We remark that Theorem 2 was proved by Evans et al. [5]. However, since g
can be localized att= 0, our method shows that Inequality (35) is sharp, i.e., the present proof shows also the sharpness of γB
c , a result of Hundertmark et
al. [10] obtained by different means.
Since — according to Tix [19] — the difference of the massive and massless Brown-Ravenhall operators is bounded, Theorem 2 shows also that the energy in the massive case is bounded from below under the same condition on γ as in the massless case.
3.2 The Jansen-Hess Energy
We now wish to treat the full relativistic energy according to Jansen and Heß as given in (28) through (31). From these equations it is obvious that the energy is positive, if the coupling constantγdoes not exceedγB
c , since the additional
energy term is non-negative. However, as can be expected, the critical coupling constant is in fact bigger, i.e. we want to prove Theorem 1 in the massless case. Lemma 2. For allu∈C∞
0 (R3)⊗C2,m= 0, andγ≤γc we have(u,Ju)≥0.
Moreover, ifγ > γc, then J is not bounded from bellow.
Proof. We write the energy density in Mellin space as given in Equations (28) through (31) as
jl,s(t) := 1−γ2(Vl(t) +Vl+2s(t)) +γ
2
8 (Vl(t)−Vl+2s(t))
2. (36)
As in the case of the Brown-Ravenhall energy we want to show thatjl,sattains its minimum forl= 0 andt= 0.
First we note, thatjl,s(t) =jl+2s,−s(t) which means that we can restrict the
following to s= 1/2, i.e., to jl,1/2.
Next we show that it is monotone decreasing inl. Forγ≤4/π we have
0≤1−γ2V0(0)≤1−γ2Vl(t)≤1−γ4Vl(t)−γ4Vl+2(t)
≤1 +γ
2Vl+1(t)−
γ
4Vl(t)−
γ
4Vl+2(t)
where use successively (64), (32), Lemma 6 in Appendix C, and the positivity of the Vl. Inequality (37) is – after multiplication by γ((Vl(t)−Vl+2(t))/2 – identical with the desired monotonicity inequality
jl+1,1/2(t)≥jl,1/2(t). (38)
For later purposes we note that functionsjl,1/2are symmetric about the origin. Next we will show that the energy density has its absolute minimum at the origin: to this end we simply show that the derivative ofj0,1/2 is nonnegative on the positive axis, ifγ≤2/(π/2 + 2/π) which is bigger than 4/π. Since
|V0(t)−V1(t)| ≤
Z ∞
0
(q0(x)−q1(x))dx
x =V0(0)−V1(0) = π
2 − 2
π
we have
−1 + γ
2(V0(t)−V1(t))≤0 (39) and obviously we have
−1−γ2(V0(t)−V1(t))≤0. (40)
Thus the derivative of the energyj0,1/2is
j′
0,1/2(t) =
γ
2[−V
′
0(t)−V
′
1(t) +
γ
2(V0(t)−V1(t))(V
′
0(t)−V
′
1(t))] =γ
2{V
′
0(t)[−1 +
γ
2(V0(t)−V1(t))] +V
′
1(t)[−1−
γ
2(V0(t)−V1(t))]} ≥0, (41) sinceV0 and V1 are symmetrically decreasing about the origin (see Appendix C).
Finally, the polynomial
j0,1/2(0) = 1−
γ
2 µπ
2 + 2
π
¶ +γ
2
8 µπ
2 − 2
π
¶2
is nonnegative forγ≤γc as defined in the hypothesis. Thus, we have
jl,s(t)≥j0,1/2(0)≥0.
4 Lower Bound on the Energy According to Jansen and Heß To distinguish the massive and the massless expressions we will indicate in this section the dependence their on the mass m by a superscript m, if it seems appropriate.
Lemma 3 (Tix [18, 20]). For all u∈C∞
0 (R3)⊗C2,m≥0, andγ≤γcB then B(u)≥m(1−γ).
Lemma 4 (Tix [19]). The expression |Bm(u)− B0(u)| is bounded for u ∈
C∞
0 (R3)⊗C2.
Lemma 5. For allm≥0 and for allu∈C∞
0 (R3)⊗C2 we have
|K˜m(u)−K˜0(u)| ≤mdkuk2 (42)
whered:=√2(12 + 25/2).
We note that the first part of Theorem 1 follows from Lemmata 2, 4, and 5. The third part is a consequence of Lemmata 3 and 5.
Proof. First we remark that
sup{|Jm(u)− J0(u)| | kuk= 1}=m sup{|J1(u)− J0(u)| | kuk= 1}.
Then it is enough to start bounding|(u,K˜1u)−(u,K˜0u)|: By the mean value theorem we have
|K˜1(p,p′
)−K˜0(p,p′
)| ≤λ|D(µ,p,p′
)| (43)
for someµ∈(0, λ) whereλ∈(0,1) is a deformation parameter andD(µ,p,p′)
is the derivative of ˜Kµ(p,p′) with respect toµ. Computing the derivative yields
|D(µ,p,p′
)|= ¯ ¯ ¯ ¯ Z
dp′′
F(µ,p,p′′ ,p′
) ¯ ¯ ¯ ¯
(44)
with
F(µ,p,p′′ ,p′
) := 1
2(2π2)2 µ∂C
∂λ(T1+...+T4) +C
∂(T1+...+T4)
∂λ
¶
(µ,p,p′′ ,p′
) (45)
where C and T1, ..., T4 are defined right below (14). Note thata(p)2 ≤1 and
b(p)2≤1/2, i.e., by the definitionT
1, ..., T4≤1/2. Furthermore we note that
∂C
∂λ =
−λ E(p′′)
1
|p−p′′|2|p′′−p′|2
µ 1
(E(p) +E(p′′))E(p)+
1
(E(p′′) +E(p′))E(p′)
¶
. (46)
First we treat∂C
∂λ(T1+...+T4). We get using the above estimates onT1through T4 and (46)
¯ ¯ ¯ ¯
∂C
∂λ(T1+...+T4)(µ,p,p ′′
,p′
) ¯ ¯ ¯ ¯≤
2
p′′
1
|p−p′′|2|p′′−p′|2 µ 1
p+p′′ +
1
p′′+p′
¶
Next we treatC∂(T1+∂λ...+T4). To this end we note
We now bound the integral operator ˜K1−K˜0 by a multiplication operator: First pickα∈R. Then we have — using the symmetry ofF(µ,p,p′′,p′) inp
where we used the Schwarz inequality in the measuredpdp′ in the last step for
fixed p′′. Now using the estimates (47) and (50) and collecting similar terms
yields
To show the above bound we break the integral into three parts
We will also use the following integral (see [13], p.124) each of the integrals in (54) do not depend on the value of p(what becomes evident after substitution of p′
→ pp′
We chooseα= 3/2 and using (55) we obtain the same bound for each integral, namely 32π4. Equation (53) proves Lemma 5 and follows by using the latter bound in (52).
A Some Useful Integral Identities
Supposef(x) =f(1/x) and supposef(x)/(1 +x) is integrable on (0,∞). Then
To show (57) we split the first integral
Z 1
abbrevi-ationql(x) :=Ql¡1
To prove this we take the integral with the complete first factor times the first summand of the second factor –we name I1– and the integral over the complete first factor times the second summand of the second factor,I2. InI1 we substitute p′′
→pp′′ whereas in I
Undoing the substitutions yields the desired result.
B The Mellin Transform
The Mellin transform is a unitary map from L2(R+) to L2(R) given by the
The Mellin convolution of two functionf andg is defined as
(f ⋆ g)(p) =
We also have
(f ⋆ g)#(s) =√2πf#(s)g#(s). (63)
C Some Properties Related to the Partial Wave Analysis of the Coulomb potential in Mellin Space
We first remark the follow property on the difference ofVl andVl+2. Lemma 6. Forl= 0,1,2, ...andt∈Rwe have Vl+2(t)< Vl(t).
Proof. From the definition ofVl in (30) we see that the claim is equivalent to ¯
¯ ¯ ¯ ¯
Γ¡l+1−it
2 ¢
Γ¡l+2−it
2 ¢
¯ ¯ ¯ ¯ ¯ 2
>
¯ ¯ ¯ ¯ ¯
Γ¡l+3−it
2 ¢
Γ¡l+4−it
2 ¢
¯ ¯ ¯ ¯ ¯ 2
.
This, however, can be easily verified using the functional equation Γ(x+ 1) =
xΓ(x) of the Gamma function in the numerator and denominator of the right hand side with x= (l+ 1−it)/2 andx= (l+ 2−it)/2.
From the definition of theVland from Formulae 8.332.2 and 8.333.3 in [7] one findsV0andV1 in terms of the hyperbolic tangent and cotangent:
V0(t) = Tg(πt/2)
t (64)
V1(t) = t
1 +t2Ctg(πt/2). (65) Moreover, both of these functions are decreasing symmetricly about the origin.
Acknowledgment: We thank Dr. Doris Jakubaßa-Amundsen for careful reading of the manuscript and pointing out several mistakes. The work has been partially supported by the European Union through its Training, Re-search, and Mobility program, grant FMRX-CT 96-0001960001, the Volkswa-gen Foundation through a cooperation grant, and the Deutsche Forschungsge-meinschaft (Schwerpunktprogramm 464 “Theorie relativistischer Effekte in der Chemie und Physik schwerer Elemente”). E. S. acknowledges partial support of the project by FONDECYT (Chile, project 2000004), CONICYT (Chile), and Fundaci´on Andes for support through a doctoral fellowship.
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Raymond Brummelhuis Birkbeck College University of London School of Economics, Mathematics and Statistics Gresse Street
London W1T 1LL United Kingdom
Heinz Siedentop Mathematik Theresienstr. 39 80333 M¨unchen Germany [email protected]
Edgardo Stockmeyer Pontificia Universidad Cat´olica de Chile Departamento de F´ısica Casilla 306
Santiago 22 Chile
